Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.5 → 9.4

Time: 11.0s

Precision: binary64

Math

\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -2.8462562866286207 \cdot 10^{-64}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\left(\frac{y}{\sqrt[3]{a - t}} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)\\ \mathbf{elif}\;a \leq 2.814499803096097 \cdot 10^{-220}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a) :precision binary64 (- (+ x y) (/ (* (- z t) y) (- a t))))

↓

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.8462562866286207e-64)
   (-
    (+ x y)
    (*
     (*
      (cbrt (/ (- z t) (* (cbrt (- a t)) (cbrt (- a t)))))
      (cbrt (/ (- z t) (* (cbrt (- a t)) (cbrt (- a t))))))
     (*
      (* (/ y (cbrt (- a t))) (cbrt (- z t)))
      (cbrt (/ 1.0 (* (cbrt (- a t)) (cbrt (- a t))))))))
   (if (<= a 2.814499803096097e-220)
     (+ x (/ (* y z) t))
     (-
      (+ x y)
      (*
       (*
        (cbrt (/ (- z t) (* (cbrt (- a t)) (cbrt (- a t)))))
        (cbrt (/ (- z t) (* (cbrt (- a t)) (cbrt (- a t))))))
       (*
        (cbrt (/ (- z t) (* (cbrt (- a t)) (cbrt (- a t)))))
        (/ y (cbrt (- a t)))))))))

C

double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - t));
}

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.8462562866286207e-64) {
		tmp = (x + y) - ((cbrt((z - t) / (cbrt(a - t) * cbrt(a - t))) * cbrt((z - t) / (cbrt(a - t) * cbrt(a - t)))) * (((y / cbrt(a - t)) * cbrt(z - t)) * cbrt(1.0 / (cbrt(a - t) * cbrt(a - t)))));
	} else if (a <= 2.814499803096097e-220) {
		tmp = x + ((y * z) / t);
	} else {
		tmp = (x + y) - ((cbrt((z - t) / (cbrt(a - t) * cbrt(a - t))) * cbrt((z - t) / (cbrt(a - t) * cbrt(a - t)))) * (cbrt((z - t) / (cbrt(a - t) * cbrt(a - t))) * (y / cbrt(a - t))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.5
Target	8.7
Herbie	9.4

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < -1.3664970889390727 \cdot 10^{-07}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -2.84625628662862075e-64

   1. Initial program 14.6

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt_binary64_16285 14.7

      \[\leadsto \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\]

   4. Applied times-frac_binary64_16259 7.6

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64_16285 7.6

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   7. Applied associate-*l*_binary64_16196 7.6

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\]

   8. Simplified 7.6

      \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \color{blue}{\left(\frac{y}{\sqrt[3]{a - t}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)}\]
   9. Using strategy rm

   10. Applied div-inv_binary64_16250 7.6

       \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\frac{y}{\sqrt[3]{a - t}} \cdot \sqrt[3]{\color{blue}{\left(z - t\right) \cdot \frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}\right)\]

   11. Applied cbrt-prod_binary64_16281 7.6

       \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\frac{y}{\sqrt[3]{a - t}} \cdot \color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)}\right)\]

   12. Applied associate-*r*_binary64_16195 7.6

       \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \color{blue}{\left(\left(\frac{y}{\sqrt[3]{a - t}} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)}\]

   if -2.84625628662862075e-64 < a < 2.81449980309609704e-220

   1. Initial program 20.9

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

   2. Taylor expanded around inf 11.0

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

   3. Simplified 11.0

      \[\leadsto \color{blue}{x + \frac{z \cdot y}{t}}\]

   if 2.81449980309609704e-220 < a

   1. Initial program 15.4

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt_binary64_16285 15.5

      \[\leadsto \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\]

   4. Applied times-frac_binary64_16259 9.7

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64_16285 9.8

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   7. Applied associate-*l*_binary64_16196 9.8

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\]

   8. Simplified 9.8

      \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \color{blue}{\left(\frac{y}{\sqrt[3]{a - t}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8462562866286207 \cdot 10^{-64}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\left(\frac{y}{\sqrt[3]{a - t}} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)\\ \mathbf{elif}\;a \leq 2.814499803096097 \cdot 10^{-220}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))